1964 AFM Relationship Between Substrate Concentration, Growth Rate and Respiration Rate of E. Coli in Continuous Culture

7/22/2019 1964 AFM Relationship Between Substrate Concentration, Growth Rate and Respiration Rate of E. Coli in Continu

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Archl y fiir Mikrobiologie 48, 1--20 1964)

Michigan State University, East Lansing, Michigan

Relationship between Substrate Concentration,Growth Rate, and Respiration Rate

of Escheric hia coli in Continuous Culture*By

I ~ A R L I ~. SI Jt IU L Z E a n d R O B E R T S . L I P EWith ll Figures in the TextRece ived December 28 , 1963)

In batch culture it is very difficult to establish a relationship between substrateconcentration and g rowth rate for a given type of microorganism. As long as thesubstrate is in ample supply growth will proceed at a constant ma ximu m rate accord-ing to the equation

x = Xoek~t 1)o r

In X/Xo) = kitand

2)1 d x ln2k 1 -- -- - - 3)x dt g

where k 1 = specific grow th rate, i.e. growth rate per un it cell weight per unit t ime.x 0 = cell conc entr atio n at t = 0.g = mean doubling time or genera tion time.

k 1 will become dependent on the substrate concentration only wh en most of thesubstrate has been consumed. At this po int the culture has practically reached itsmaximum density and the remaining substrate concentration changes rapidly,leaving no time for the measu remen t of grow th rates at any specific subs trat e level.

This difficulty can be overcome by th e use of the continuous flow culture und ercomplete mixing conditions. Equa tion s defining the cell concentratio n and the sub-strate concentration in the reactor as well as the specific growth rate at steady statecondi tions hav e been developed by Mo~oD 1950), N ow cK and SZILA~D 1950) an dH ~ mm T , ELSWO~TH and T~LLI~G 1956). The equati ons sta te th at

k l = / I v = I ) 4 )

where lq = specific grow th rat e] = flow rate of nut rie nt solutionv = volume of reacto rD = dilut ion rat e = inverse of dete ntio n time

* Material contain ed in this p aper was s ubm itt ed as a thesis in parti al fulfillmentof the requi remen ts for the Ph .D . degree of Dr. R. S. LIVE.

Arch . Nikrobiol. Bd. 48 1


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E ~ L L SCHU LZE and ROBEI~T S LIPE:

a~ = c e l l c o n c e n t r a t i o n i n r e a c t o r o r e f f l u e n t a t s t e a d y s t a t e c o n d i t i o n s= s u b s t r a t e c o n c e n t r a t i o n i n re a c t o r o r e f f l u e n t a t s t e a d y s t a t e c o n -d i t i o n sY ~ y i e l d f a c t o r = u n i t w e i g h t o f c e ll s f o r m e d p e r u n i t w e i g h t o f s u b s t r a t euseS = s u b s t r a t e c o n c e n t r a t i o n i n f e e d s o l u t io nS , = s u b s t r a t e c o n c e n t r a t i o n a t w h i c h k i = k , ~ / 2k~, = m a x i m u m g r o w t h r a t e fo r t h e g i v e n s e t o f c o n d i ti o n s .

F r o m t h e s e e q u a t i o n s i t fo l lo w s t h a t u n d e r s t e a d y s t a t e c o n d i t io n s1 . t h e c e l l c o n c e n t r a t i o n i n t h e r e a c t o r i s d e t e r m i n e d b y t h e y i e l d f a c t o r Y a n dt h e d i ff e re n c e b e t w e e n i n g o i n g a n d o u t g o i n g s u b s t r a t e c o n c e n t r a t i o n ( e q u a t i o n 5 ).T h e i n v e s t i g a t o r c a n t h e r e f o r e c h o o s e t h e c e ll c o n c e n t r a t io n a t w h i c h h e w a n t s t oo p e r a t e m e r e l y b y v a r y i n g t h e s u b s t r a t e c o n c e n t r a t i o n i n t h e f e e d s o lu t io n . T h i s i si m p o r t a n t b e c a u s e a t h i g h ee l1 c o n c e n t r a t i o n s t h e o x y g e n s u p p l y m a y b e c o m e al i m i t i n g f a c t o r f o r a e r o b i c c u l t u r e s .2 . th e g r o w t h r a t e i s e q u a l to t h e d i l u t i o n r a t e a s lo n g a s s t e a d y s t a t e c o n d i t i o n sa r e m a i n t a i n e d ( e q u a t io n 4 ). S in c e t h e v o l u m e o f th e r e a c t o r i s c o n s t a n t t h e g r o w t h r a t ei s d e t e r m i n e d b y t h e f e e d r a t e a n d c a n b e v a r i e d s i m p l y b y v a r y i n g t h e f e e d r a t e .T h i s p r o v i d e s a c o n v e n i e n t m e a n s o f e s ta b l is h i n g t h e r e l a t io n s h i p b e t w e e n s u b s t r a t ec o n c e n t r a t i o n a n d k 1. D e c r e a s i n g t h e f e e d r a t e r e s u l t s i n a d e c r e a s e o f D = k i a n dt h i s p r o d u c e s a d e c r e a s e i n s u b s t r a te c o n c e n t r a t io n d u e t o t h e i n c r e a s e d c o n t a c tt i m e . C o r r es p o n d in g t o a g i v e n D o r k i v a l u e t h e c u l t u r e w i l l t h e r e f o re a u t o m a t i c a l l ye s t a b li s h a d e f i n it e s u b s t ra t e c o n c e n t r a t i o n i n t h e r e a c t o r a s l o n g a s t h e g i v e n s e t o fc o n d i ti o n s re m a i n s u n c h a n g e d . T h e s y s t e m c a n t h u s b e o p e r a t e d a t a s e r i es o f s t e a d ys t a t e l o w s u b s t r a t e c o n c e n t r a t i o n s , i . e . i n t h e r a n g e w h e r e k 1 i s d e p e n d e n t o n S .T h e b a s i c b a l a n c e e q u a t i o n f o r t h e r e a c t o r i s :

d x / d t = k l x - - D x . (7)O b v i o u s l y D c a n n o t e x c e e d th e v a l u e o f k l i f s t e a d y s t a t e i s t o b e m a i n t a i n e d . A tD > k ~ , d x / d t i s n e g a t i v e a n d t h e c e ll c o n c e n t r a t i o n i n t h e r e a c t o r w i l l t h e r e f o r ed e c re a s e p r o g r e s si v e l y w i t h t i m e u n t i l e v e n t u a l l y n o c e ll s a r e l e ft . T h i s p r o v i d e s am e a n s o f e s t a b l i s h i n g k ~ f r o m c o n t i n u o u s f l ow d a t a . O n e h a s o n l y t o i n c r e a s e D t ot h e p o i n t w h e r e t h e c e ll c o n c e n t r a t i o n i n t h e r e a c t o r b e g i n s t o s h o w a s t e a d y d e c li n e .

T h e b a s i c e q u a t i o n r e l a t i n g s u b s t r a t e c o n c e n t r a t i o n a n d k s w h i c h h a sb e e n e m p l o y e d t o d e v e l o p e q u a t i o n (5 ) a n d (6 ) i s :

T h i s i s f o r m a l l y e q u a l t o a n a d s o r p t i o n i s o t h e r m o r t o t h e M i c h a e l i s -M e n t e n e q u a t i o n w h i c h i s u s e d i n e n z y m e k i n e t i c s . T h e b a c t e r i a l c e l l i st h u s c o n s i d e r e d a s a n e n z y m e m o l e c u l e r e a c t i n g w i t h t h e s u b s t r a t e S .S ~ c o r r e s p o n d s t o t h e M i e h a e l i s - M e n t e n c o n s t a n t a n d r e p r e s e n t s t h e s u b -s t r a t e c o n c e n t r a t i o n a t w h i c h t h e r e a c t i o n v e l o c i t y , i .e . t h e g r o w t h r a t ek 1 r e a c h e s i / 2 o f i t s m a x i m u m v a l u e . T h e c u r v e i s h y p e r b o l i c a n d k m w o u l db e r e a c h e d a t i n f i n i t e s u b s t r a t e c o n c e n t r a t i o n . T E IS S IE R ( 1 93 6 ) p r o p o s e da d i f f e r e n t t y p e o f e q u a t i o n

d k i / d S = c k ,~ - - k l ) (9)o r i n t e g r t e dk ~ = k , . 1 - e - c a ) . 1 0 )


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Substrat~ concentration growth and respiration rate in continuous culture 3T h i s e x p r e s s i o n i s f o r m a l l y s i m i l a r t o t h e e q u a t i o n o f a f i r s t o r d e r r e a c t i o nw h e r e ] c i n c r e a s e s w i t h i n c r e a s i n g s u b s t r a t e c o n c e n t r a t i o n p r o p o r t i o n a l l yt o k m k ~ E x p e r i m e n t a l c u r v e s d e m o n s t r a t i n g t h e r e l a t i o n s h i p b e t w e e ng r o w t h r a t e a n d s u b s t r a t e c o n c e n t r a t i o n h a v e b e e n p u b l i s h e d b y s e v e r a la u t h o r s a n d f o r s e v e r a l g r o w t h l i m i t i n g f a c t o r s s u c h a s g l u c o s e , m a n n i t o la n d l a c t o s e M O ~ O D 1 9 4 2 , 1 9 4 9 ) , t r y p t o p h a n N o v l c K 1 9 5 5 ) a n d p h o s -p h a t e S c H u L Z E 1 9 5 6 ) . A l l t h e s e c u r v e s a r e o f t h e s a m e g e n e r a l s h a p ew h e r e t h e g r o w t h r a t e a s y m p t o t i c a l l y a p p r o a c h e s a m a x i m u m v a l u e w i t hi n c r e a s i n g c o n c e n t r a t i o n o f t h e l i m i t i n g f a c t o r . E i t h e r o n e o f t h e e q u a t i o n s(8) or (10) could represent these curves. So far a decision between the twoformula t ions has not been made probab ly due to the fac t tha t the exper i -menta l da ta were insuffic ient . Par t ly because of the reason jus t ment ione dand par t l y because the Michae l is -Menten equat i on is more convenient , thedeve lopme nt of the cont inuous f low equat ions has been based exc lus ive lyon equation (8).

In the fol lowing an a t tem pt is made to tes t the va l idi ty of equat ions(8) and (10) by num erica l analysis of exp erim ent al data . As a consequen cenew s teady s ta te equ at ions def ining ce ll - and subs t ra te concentra t ionbased on equ atio n (10) will be developed. In a ddit i on the exp erim ent alda ta wil l be used to es tabl ish mat hema t ica l re la t ionships be tween growthra te , oxygen consumption ra te and the ra te of subs t ra te removal .

aterials and etho dsIn the experimental apparatus two different pyrex reaction chambers were used;one with a constant culture volume of 1 1 for the higher feed rates and the other witha volume of 31 for the very small feed rates. The temperature of the reaction chamberwas maintained at 30 C 0.5 C by means of a constant temperature wa ter bath.The system used for providing the reaction chamber with a constant supply ofsubstrate is labelled system A in Fig. 1. The feed solution was pumped from a 201storage bottle by means of a Sigmamotor Model T65 pump. By closing clamp A in

System A the flow of the feed solution was directed down into a burette and thefeed rate was determined as ml/min with the a id of a stop watch.The system used for the aeration of the culture in the reaction chamber is shownas System B in Fig. 1. Compressed air controlled by a pressure regula ting valve setat 51,6 mm Hg was passed through a We t Test meter in order to measure the flow rateand then through a large sterile cotton filter before entering the reac tion camber. Twofrit ted glass diffusors were used as aeration devices. The dissolved oxygen rangeobtained during the experiments was 0.5--4.8 ppm, in other words, the dissolvedoxygen concentration did not decrease below 0.5 rag/1 at any time. The rate of airsupply, varied from 1.01 per liter reactor volume per rain at small D values to 3.1 1per liter reactor volume per rain at the higher D values. The air passing through theunit no t only served as an oxygen supply but also as a mixing device to thoroughlymix the incoming substrate with the cell suspension in the reaction chamber.Effluent from the react ion chamber was discharged through a 1/4'+ overflow tubewhich also served as the air exit line.The system for the sampling of culture from the reaction chamber is shown assystem C in Fig. 1. In order to obtain a sample c lamp B was closed, clamp C was

1


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K A r L L . S C~ VL Z ]~ a n d R O B E R T S . L I F E :o p e n e d a n d t h e l i q u i d s a m p l e d r a w n i n t o a s a m p l e c o n t a i n e r b y m e a n s o f a v a c u u m .C l a m p C w a s th e n d o s e d , d a m p B o p e n e d a n d t h e l i q u i d r e m a i n in g i n s y s te m C w a sr e t u r n e d t o t h e r e a c t i o n c h a m b e r .A l l p a r t s o f t h i s a p p a r a t u s w e r e s t e r i l i z e d i n a n a u t o c l a v e a t 1 5 l b s. p r e s s u r e f o ra t l e a s t 3 0 m i n . T h e r e a c t i o n c h a m b e r , a e r a t i o n s y s t e m , s a m p l i n g s y s t e m , a n d f e e ds y s t e m w e r e s te r i li z e d t o g e t h e r . T h e o t h e r p a r t s o f t h e a p p a r a t u s w e r e s t e ri l iz e ds e p a r a t e l y a n d c o n n e c t e d t o t h e r e s t o f t h e s y s t e m b y m e a n s o f t h r e e - in c h p i e c e s o fs t er il e , h e a v y r u b b e r t u b i n g .T h e c o m p o s i t i o n o f t h e n u t r i e n t s o l u t i o n w a s a s f o l lo w s ( g /l ) : G l u c o s e 1 . 0 ; U r e a0 . 5 ; K H 2 P O 4 0 . 1 4; M g S O 4 0 . 0 3 ; F e S 0 4 0 . 0 0 5 ; C aC 12 0 . 0 1 ; Y e a s t e x t r a c t 0 . 0 1 .T h e px~ w a s a d j u s t e d t o 7 . 0 b y a d d i n g 1 00 m l o f 0 .1 N s u lf u r i c a c i d t o e a c h 2 0 1o f th e m e d i u m .

o ~ D J em B 8 /g, Y j /s l e m S 9 l f I .

/ 2 I I

7Fig. 1. Schematic diagr am of continuous flow unit. Syst em A: nutrien~ supply, I pum p, 2 gradu ate dbure tte , 3 clamp. System B: air supply, g Wet-Te st meter, 5 air filter, 6 needle valve, 7 diffuser.,System C: sampling, 8 and 9 clamps, 1 and 11 air filters, 12 two- way stopcock, 13 samDling tube,14 growth chamber, 15 water bat h, 16 overflow line

T h e a m o u n t o f e a c h c o n s t i t u e n t i n t h e m e d i u m w a s s u c h t h a t n i t r o g e n , p h o s p h o -r u s , m a g n e s i u m , i r o n , c a l c iu m a n d n u t r f l it e s ( s u p p li e d b y t h e y e a s t e x t r a c t ) w e r ep r e s e n t i n e x ce s s. T h e i n t e n t i o n w a s t o m a k e g l uc o s e t h e o n l y g r o w t h l i m i t i n g f a c t o r.I n i t i a l l y s e v e r a l s t r a in s o f b a c t e r i a i s o l a t e d f r o m s e w a g e w e r e te s t e d . H o w e v e r

t h e s e w e r e f o u n d u n s u i t a b l e b e c a u s e o f c l u m p i n g , s t ic k i n g t o t h e s id e s o f t h e r e a c t i o nc h a m b e r , i n c o n s i s te n t g r o w t h o r a t e n d e n c y t o s e t t l e . T h e o r g a n i s m f i n a l ly s el e c te dw a s a s t r a i n o f scherichia cell o b t a i n e d f r o m D r . E . D . D E VE ~E U X, D e p a r t m e n to f M i c r o b i o l o g y a n d P u b l i c H e a l t h , M i c h i g a n S t a t e U n i v e r s i t y . A f t e r a s h o r t a d a p -t a t i o n p e r i o d t h i s o r g a n i s m w a s f o u n d t o b e w e l l s u i t e d f o r u s e in t h e c o n t i n u o u sf lo w s y s te m . I t s g r o w t h w a s e v e n l y d is p e r se d w i t h n o t e n d e n c y t o c l u m p o r t o s t i c k

t o t h e s i d e s o f t h e r e a c t i o n c h a m b e r .T h e f o l l o w i n g a n a l y t i c a l d e t e r m i n a t i o n s w e r e m a d e d a i l y d u r i n g t h e o p e r a t i o no f t h e c o n t i n u o u s f l o w u n i t :1 . G l u c o se - - b y a p h o t o m e t r i c a d a p t a t i o n o f t h e S o m o g y i m e t h o d a s g i v e n b yNELSO N (1944) .

2 . C e l l c o n c e n t r a t i o n - - f r o m a c a l i b r a t i o n c u r v e r e l a t i n g o p t i c a l d e n s i t y a n dr ag /1 c e l l d r y w e i g h t a t a w a v e l e n g t h o f 35 0 m .

3 . D i s s o l v e d o x y g e n - - a c c o r d i n g to t h e s t a n d a r d W i n l f le r m e t h o d ( S t a n d a r dM e t h o d s f o r t h e E x a m i n a t i o n o f W a t e r , S e w a g e a n d ~ n d u s t r ia l W a s t e s , 1 95 5). 5 0 m lo f sa m p l e f r o m t h e r e a c t o r v e s s e l w e r e u s e d f o r e a c h d e t e r m i n a t i o n .


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Substrate concentration growth and respiration rate in continuous culture 54. Oxygen uptake rates in a circular 20 uni t Warburg apparatus using thestandard procedure recommended by UMBREIT eta] . (1957). 5 ml samples of the cell

suspension from the reactor were placed in 6 Warburg flasks which were operated inparallel and three manometer readings were taken at 5 or 10 min intervals. For mostD-values this procedure was repeated on 2 or 3 successive days.5. p~ -- PH determinations were made three or four times daily using a Beck-man Model H-2 pH-meter. Due %o the production of acid by the organisms it wasnecessary to adjust the PH of the culture by adding sterile 0.1 N sodium hydroxidesolution three to four times daily from a 500 ml bottle connected to the reactor. Thisprocedure maintained the pH in the reactor between 6.4 and 7.0.Contaminationwas checked b y frequent microscopic examination of wet mountsand Gram stained preparations of the reactor cell suspension under the oil immersionlens. When contaminationdid occur the uni t was shut down, cleaned, sterilized, andinoculated again with a pure stock culture.On the bacterial cell material total nitrogen was determined according to theA.0.A.C. Official I~Iethods of Analysis (1950). Ash determinat ions were made in anelectric muffle furnace heated to a low red heat (650~ Viable cell counts wereobtained by a drop-plate procedure reported by M~LMANN and BROIT~AN (1956).This procedure is based on the premise that colonies will appear on a suitablemedium from each culturable cell placed on the medium.

esultsValues for /% were first established from aerat ed batc h cultures by

plot t ing log cell densi ty versus t ime as shown in Fig. 2. The ini t ial sub-strate con cen trat i on was 980 mg/1 glucose, so th at subst rata was not alimi tin g factor. The first setof dat a (Flask A) was obt aine d 7from the original culture ofE coli used in s tar t ing the .~ 8con tinuo us f low experiments.For the second set of data ~Y(Flask B) cell mat eri al whic hhad been mai ntai ned in con- -~tin uou s f low culture for a periodof 3 mo nt hs was used as in- _g 3oculu m. This served as a checkagainst the possibil i ty of mu-tat iona l changes in the growthrate of the E coli s t rain dur ingcontinuous f low culture. Theslopes of the lines were com-puted by the metho d of leas t

I x ~ o

215 .z,,~----8

2 I [ I I Ior~ 'A 2 3 r ,5- 8 7//o's-,.('8 / 2 3 g 5f/me hou~:Fig 2 ]~atch culture dete rmin atio n of kin. A Beforeuse in contin uous flow culture ; B After use in

continuous flow culture

squares a nd resul ted in k m = 0.67 an d 0.69 for Flask A an d B respective ly.A stat is t ical t -test compa ring the two slopes showed th at the differencebet wee n the slopes was not sig nifica nt; t = 3.008 as compa red to 3.335at the 1 ~ pro ba bil ity level for 8 degrees of freedom.


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KAI~L L S C H U L Z E a n d R O B E R T S LIPE:I n e x p e r i m e n t N o . i t h e c o n t i n u o u s f l o w u n i t w a s o p e r a t e d a t a s e r i e s

o f D v a l u e s r a n g i n g f r o m 0 . 0 6 t o 0 . 8 5 p e r h o u r b y v a r y i n g t h e f e e d r a t e .E a c h f e e d r a t e e x c e p t a t t h e h i g h e s t a n d l o w e s t l e v e l w a s m a i n t a i n e d f o r3 d a y s t o o b t a i n s t e a d y s t a t e c o n d i t i o n s . A v e r a g e d a t a f o r c e l l a n d s u bs t r a t e c o n c e n t r a t i o n i n t h e r e a c t o r a n d f o r t h e o x y g e n u p t a k e r a t e s a r el i s t e d i n T a b l e i . I n e x p e r i m e n t N o . 2 t h e t e s t w a s r e p e a t e d w i t h D v a l u e sr a n g i n g f r o m 0 . 0 6 t o 0 . 7 3 a s s h o w n i n T a b l e 2 . T h e t a b l e s d e m o n s t r a t e

Dhr-10.0590.0910.1240.1770.2410.3020.3580.4250.4850.5460.6100.6620.7250.7920.852

Table 1. Experiment No 1Avg. substrate cone. in feed = 964 rag/1 glucose, temp, 30~

cone Qo~3 m g/g y ie ld fac tor mg/gkce l l wtellsrag/1 rag/1 cell wt/hr Y per hour5.18.313.320.330.437.043.158.0

74.596.5112.0161.0195.0266.0386.0

422429431428421433420420413433426434396311156 *

48.674.3100.0150.0202.0269.0322.0356.0379.0448.0478.05 0 5 . 0487.0500.0490.0

0.4400.4490.4530.4530.4510.4670 4550.4640.4650.499O.5000.5400.515

1342032743905346477859171,0431,0911,2201,225i ,420

* after 20 hrs operation.** after 12 hrs operation.1 avg. from 6 determinations.2 avg. from 3 determinat ions.8 avg. from 36--48 determinations.

~glucoseoxidized34.035.034.335.535.839.538.036.233.838.237.039.032.4

th at s teady s tate condi t ions were maintMn ed up to a D-vMue of abou t0.79. At higher D-values the cell con cen trat ion in the reac tor b egan todecline continuous ly. This leads to the conclusion th at the value of kmshould be close to 0.79 un der cont inuo us f low condit ions. This est imatefor k~ is s igni f icant ly higher th an tha t found f rom batch cul ture data. Ashas bee n poin ted out b y EE~BE~T et al. (1956) the re ason for thisdifference is proba bly the fact th at the actual wash-out rate of cells is lessth an t ha t given by equat ion (7). Equ at i on (7) i s based on the a ssumpt io no f i n s t a n t a n e o u s c o m p l e t e m i x i n g i n t h e r e a c t o r . I t i s r e a s o n a b l e t o e x p e c tt h a t t h e e x p e r i m e n t a l a p p a r a t u s H I o n l y a p p r o a c h t h e s e t h e o r e t i c a l c o nd i t i o n s .


7/20

Substrate concentration growth and respiration rate in continuous culture 7A plot showing the relat ionship be tween growth rate/c 1 and substra te

con cen trat ion S for ru n ~qo. 1 is given in Fig. 3. The curve demo nstrat estha t ]c increased wi th increas ing subs t rate c oncentra t ion and asympto-t ical ly ~pproached a max im um value at a subs t rate concen trat ion of

Tuble 2. Experiment No. 2Avg. substrate cone. in feed968 mg/l glucose, temp. 30~D = k~ S cell cone.hr ~ rag/1 rag/10.060.120.240.310.430.530.600.660.69:0.710.73

6.013.033.040.064.0102.0122.0153.0170.0221.0210.0

427434417438422427434422430390352

0 . 5 - -

g .8

0 .~

0 . 2

~B o

~ ~//0 80 120 1 0 ~00

Fig. 3. Relationshipbetweensubstrate concentrationand growth rate. Comparisonbetween epxerimentaldata and theoretical curves, o experimentaldata,exp. No. 1

about 220 mg/1 glucose. ]3elow this level kl was dependent on substrateconc entra t ion. To test the v ali di ty of equa tions (8) and (10) the followingway was chosen.

Equation (8) can be rearranged to1/,~1 = 1I/~,,, + s: I /~ , , , x 11 3 (11)

This is the eq uat ion of a strMght Yme, y = a bx, where the y-interceptwou ld be equal to 1//c~, the x-in terc ept = -- 1/S~ and the slope b equal toS~ / I~ . The curve obta ined from plott ing 1/k 1 versus 1/S from the dat a,of ru n No. 1 in this ma nn er is shown in Fig. 4. This is equiv ale nt to theLineweaver-Burke plot Dixonan d WEBB I958) used in en zym e kinetics.To obt ain the l ine of best f it the co nstan ts a = 1/k~ and b = S~/k,~ werecalculate d by the least squares me tho d result in g in /c~ = 0.92 a ndN~ ~ 73. The same pro cedure was followed usin g data from r un No. 2with the results k~ = 1.05 an d S~ = 99. The relatio nshi p bet wee n ]c an d

according to equ atio n (8) is therefore g iven by

orkl = 0.92 ( 7 - ~ - f f ) (12)

kl= 1.05 9 ~ ) 12a)


8/20

KARL L. Sc~uLz]~ and ROBERT S. LIFE:I n t e s t i n g e q u a t i o n (10) th e p r o b l e m a g a i n w a s to d e t e r m i n e v a l u e s

f o r k ~ a n d c f r o m a s e t o f e x p e r i m e n t a l d a t a . S e v e r a l r a t h e r i n v o l v e dm e t h o d s t o o b t a i n t h e s e c o n s t a n t s h a v e b e e n d e s c r i b e d ( R E E D a n dTHER~AVLT 1931; T~O~AS 1937; ~OOR E et al . 1950). On e o f the s im ple rp roced u res i s t he m e t h od o f ave rag es (HO]~LSC~[E~, A ~O LD an d PI ]~o ]~1952) . Equa t i on (9 ) can be r ewr i t t en a s

dI ql d,S : c lc~ -- c k 1 (13)Th i s i s t he equa t i o n o f a s t r a i gh t l ine y : a - - c x where

y ~ dI ~l / dS ; a ' ~ c k~ ; x = 1cl; c ~ constant.The y - i n t e rcep t i s equa l t o a ' ~ c k ~ , t he x - i n t e rce p t ~ - k~ and t he s l ope: c. A pl ot o f L l k l / A S v e r s u s k l i s s h o w n i n F i g . 5 , d e m o n s t r a t i n g t h a t t h e

/ 0

/ 2

-/ / / o I r I i I i I iq 8 /2 /G 20l/ ~X iO8]Rig. 4. 1 lot of 1]k, ~ersus 1/s

0. 0120.010 o

0.0081

I.OOg0 0 0 2 f ,

0 0.2 0.// O.B 0.8k s k~ I

Fig 5 Plot of Ak~ AS ersus k~

exp e r i m en t a l da t a f i t a s t r a i gh t l i ne r eas ona b l y we ll . The l i ne o f bes t f i ta s ob t a i n ed b y t he m e t ho d o f ave rage s r e su l t ed i n km ~ 0 .76 and c ----0 .014fo r run N o . I an d k m = 0 .77 an d c ---- 0 .012 fo r ru n N o . 2. The r e l a t ion -sh i p be t wee n k 1 and S acco rd i ng t o equ a t i on (10) i s t he re fo re g i ven by

~l = 0.76 (1 - - e-~176~) (14)or

/c1 = 0.77 (1 -- e ~176 ~) (14a)~ o r c o m p a r i s o n w i t h t h e e x p e r i m e n t a l d a t a f r o m r u n N o . 1 c u r v e sem p l oy i ng equa t i ons (12) and (14) were p l o t t ed a s shown i n F i g . 3 . Bo t hc u r v e s f it t h e e x p e r i m e n t a l d a t a . I n g e n e r a l c u r v e B r e p r e s e n t i n gequ a t i on (14) shows a c lo se r ag r eem en t w i t h t h e exp e r i m en t a l re su l t s .T h e m a i n p o i n t h o w e v e r is t h a t t h e n u m e r i c a l a n a l y s is o f a s e t o f e x p e r i -


9/20

Substrate concentration growth and respiration rate in continuous culture 9mental data using equation (8) resulted in values for the constants k~ andS~ which could not be verified by the experiment. At a D value of 0.85(Table 1) no steady state was possible, indicating that k,~ values of 0.92or 1.05 were considerably out of range. The same was tru e for the S~values. According to equat ion (8), S n represents the subst rate concen-tration at which k1 reaches 1/2 of km. The experimental data show thatthis po int should correspond to about 50 mg/1 glucose, whereas equations(12) and (12a) provide values of 73 and 99 mg/1.

In contrast the kw values of 0.76 and 0.77 obtained from a numericalanalysis of the dat a according to eq uati on (10) agree well with a criticaldilution r ate near 0.79 as indicated by the loss of steady s tate conditions.Furt herm ore equation (10) produces a value of S- = 49 for k I = k w / 2which is in agreement with the experimental data.

According to the boundary conditions equation (5) is not valid forD > kw. However as menti oned previous ly the net rate of increase or lossof cells from the reactor under complete mixing conditions is given byequation (7). Under the reasonable assumption that at D > k~ the cellscontinue to grow at a rate equal to kw we ob tain:

dx N t = k,~ - D) x (15a)The portion of cells remaining in the reactor after time t will then be

x t / x~ = e ~m - -b ) t (16)where x, = cell concentration in reactor after time t, rag/l; x~ = initial cell concen-tration, rag/l; t ~ time, hours.The equati on shows tha t the rate at which the cell concentr ation decreasesin the reactor depends on how much D exceeds kw.

In experiment No. 1 the system wasoperated at D = 0.85 for 12 hrs star ting at 3 0 0an initial cell con centrati on of 311 mg/J at ~.t = 0. Checks on the cell con cen tra tion were E 20omade 2.7, 7 and 12 hrs after D had been ~increased to 0.85. Using equa tion (16) theo- ~ /00rctical cell concentrations were calculatedfor k~ = 0.76. The results as shown inFig.6 demonst rate a fair degree ofagreement between the theoretical curveand the actual decrease in cell concen-tration.

- - o ' '~ ~..,~

r r Jz 8 2?)me, hours

F i g . 6 . W a s h - o u t o f c e l ls , D ~ 0 . 8 5 ,km 0 . 7 6 , e x p e r i m e n t N o . 1 .

. . . . t h e o r e t i c a l c u r v e ;o e x p e r i m e n t a l d a t a

T h e y i e l d / a c t o rThe yield facto r Y is given by th e rat io of the weight of bacterial cells

produced to the weight of substrate consumed per unit time. The con-tinuous flow technique provides a very convenient means of establishing


10/20

10 KA rL L. SCHUnZV. and ROBERT S. L~ ]~:Y a t a s e ri e s o f v a r y i n g g r o w t h r a t e s k s. T h e a m o u n t o f s u b s t r a t e c o n -s u m e d is g i v e n b y ( S - - S ) t i m e s m l f e e d p e r h o u r a n d t h e a m o u n t o f c e ll sp r o d u c e d i s g i v e n b y t h e c e ll c o n c e n t r a t i o n 2 t i m e s m l f e e d p e r h o u r . T h ef lo w r a t e s c a n c e l o u t a n d

sz,._ s -- -- ~ ( t7)T h e Y - v a l u e s o b t a i n e d f o r r u n N o . 1 a r e l i s t e d i n c o l u m n 5 o f T a b l e 1.T h e d a t a s h o w t h a t Y in c r e a s e d f r o m 0 . 44 a t a k 1 v a l u e o f 0 . 06 t o 0 .5 4 a ta k 1 v a l u e o f 0 .6 6 . T h i s m e a n s t h a t a t l o w g r o w t h r a t e s 1 / ] ( ~ 2 . 27 gg l uc o s e w e r e c o n s u m e d i n t h e p r o d u c t i o n o f i g c e ll m a t e r i a l w h e r e a s a th i g h g r o w t h r a t e s o n l y 1 / Y --~ 1 .8 5 g g lu c o s e w e r e c o n s u m e d p e r g r a m c e llw e i g h t p r o d u c e d . I t a p p e a r s t h a t a t h i gh g r o w t h r a t e s s u b s t r a t e w a s m o r ee f f ic i e n tl y c o n v e r t e d t o c e ll m a t e r i a l . T h i s m a y b e r e l a t e d t o t h e p h e n o -m e n o n o f e n d o g e n o u s r e s p i r a t i o n . A s i m i l a r in c r e a s e o f t h e y i e l d f a c t o rh a s b e e n d e m o n s t r a t e d b y I ~ E ~ B ~ T ( 19 58 a ) f o r Aero bacter aerogenes a n df o r Torula ut i l i s

Relat ionship between k 1 and respirat ion rateT h e o x y g e n c o n s u m p t i o n r a t e s o b t a i n e d f r o m W a r b u r g m e a s u r e m e n t s

f o r a s e r ie s o f D v a l u e s i n r u n N o . 1 a r e l i s t e d in c o l u m n 4 o f T a b l e 1 a n da p l o t o f Q o , v e r s u s D i s s h o w n i n F i g . 7 . S i n c e i t w a s p r e v i o u s l y d e m o n -s t r a t e d t h a t D ~ k 1 f o r v a l u e s u p

5

~zoo

3002

/00

_ o

02 0 ~ 0 8 8O=kz hm~

Nig.7. Relationship between growth rate kland respiration rateQo, = 16 ~- 770 kl

a n dks_ _ Qo~ - 16770

t o 0 .7 9 t h e c u r v e s h o w s t h a t t h er e s p i r a t i o n r a t e w a s d i r e c t l y p r o -p o r t i o n a l t o t h e g r o w t h r a t e k 1.A s i m i l a r r e s u l t w a s o b t a i n e d b yH E R B ] ~ T ( 1 9 5 8 b ) f o r A aerogenesT h e c u r v e f o ll o w s t h e e q u a t i o n

Qo~ = b + d k1 (18)where Qo~ = oxygen consum ption rate,mg OUg cel l wt lhr ; b = y-intercep t , re-prese nting Qo~ at k1 = O; d = slop e.U s i n g t h e l ea s t s q u a re s m e t h o dv al u e s of b = 16 a n d d -----7 7 0 w er eo b t a i n e d f r o m t h e e x p e r i m e n t a ld a t a s o t h a t

(19)

20)I n g e n e ra l t h e d a t a d e m o n s t r a t e t h a t t h e r e s p i r a t i o n r a t e o f b a c t e r i a lc e l l s c a n v a r y o v e r a l a r g e s c a le f r o m 1 6 m g 0 2 p e r g r a m c e l l w e i g h t p e r


11/20

Sub stra te concentra t ion grow th and respira t ion ra te in cont inuou s cul ture 11h o u r a t k 1 = 0 , i .e . w h e n t h e c e ll s a r e n o t g r o w i n g , t o a b o u t 5 0 0 m g 0 3p e r g r a m c ell w e i g h t p e r h o u r a t t h e m a x i m u m g r o w t h r a t e . I s o l a t e dd e t e r m i n a t i o n s o f t h e r e s p i r a t i o n r a t e i n b a c t e r i a l c u l tu r e s t h e r e f o r e d on o t m e a n m u c h , s in c e t h e r a t e d e p e n d s o n t h e c o n d i t io n s o f t h e c u l t u ree x i s ti n g a t t h e t i m e th e m e a s u r e m e n t w a s m a d e . T h u s t h e r e s p i r a t io n r a t eo f 3 9 0 m g 0 2 p e r g r a m c e ll w e i g h t p e r h o u r a t 3 2 ~ g i v e n b y S PE OT O ~(1956) fo r E co l i w o u l d c o r r e s p o n d to a g r o w t h r a t e o f 0 .4 9 u n d e r t h ee x p e r i m e n t a l c o n d i t io n s d e s c r i b e d h e r e . I t i s b e l i e v e d t h a t Q o , --- 1 6r e p r e s e n t s t h e e n d o g e n o u s r e s p i r a ti o n r a t e o f t h e s t r a i n o f E eo l i u s e di n t h e e x p e r i m e n t s d e s c r i b e d h e r e . A p p a r e n t l y t h e r e e x i s t c o n s i d e r a b l ev a r i a t i o n s i n e n d o g e n o u s r e s p i r a t i o n r a t e s f o r d if f e re n t s p e c ie s o f b a c t e r i a .T h e v a l u e d e d u c e d f r o m t h e c u r v e s h o w n b y I ~E R ] ~ E ~ ( 19 5 8 b ) f o r A .aerogenes i s b ~ 1 00 . F r o m t h i s i t a p p e a r s t h a t A a e r o g e n e s h a s a n e x t r e m e -l y h ig h e n d o g e n o u s r e s p i ra t i o n r a t e . T h e c u r v e s h o w n i n F i g . 7 f u r t h e rd e m o n s t r a t e s t h a t Q o , r e a c h e d i t s m a x i m u m v a l u e a t D ~ kr a n d t h a tQ o , w a s i n d e p e n d e n t o f D a t D v a l u e s l a rg e r t h a n k ~ . T h i s s u p p o r t s t h ec o n c e p t t h a t t h e c el ls c o n t i n u e d t o g r o w a t a r a t e e q u a l t o k ~ w h e n Dw a s l a r g e r t h a n t h e c r i t i c a l d i l u t i o n r a t e D c I n a d d i t i o n e q u a t i o n (2 0)s h o w s t h a t i t is p o s s ib l e t o c o m p u t e g r o w t h r a t e s f r o m r e s p i r a t io n r a t e si f t h e e n d o g e n o u s r a t e a n d t h e s lo p e ar e k n o w n .

R a t e o / s u b s tr a t e u p t a k eT h e r a t e o f s u b s t r a t e c o n s u m p t i o n p e r g r a m c ell w e i g h t p e r h o u r c a n

e a s il y b e c a l c u l a te d f r o m c o n t i n u o u s f lo w d a t a :~ 2 - / s - / 8 ( 2 1 )v 2

w h e r e k~ ---- s p e ci fi c r a t e o f g lu c o s e u p t a k e i n m g p e r g r a m c el l w e i g h t p e rh o u r . T h e r e s u l t i n g v a l u e s a r e s h o w n i n c o l u m n 6 o f T a b l e 1 . T h e a c t u a lo x y g e n c o n s u m p t i o n c a n t h e n b e c o m p a r e d w i t h t h a t r e q u ir e d f o r c o m -p l e t e o x i d a t i o n o f t h e g l uc o se c o n s u m e d b y t h e c ells . T h i s c o m p a r i s o ns h o u l d p r o v i d e a n e s t i m a t e o f t h e f r a c t io n o r p e r c e n t a g e o f g l uc o se w h i c hw a s o x id iz e d . T h e d a t a l is t e d i n c o l u m n 7 o f T a b l e 1 s h o w t h a t t h e s ev a l u e s v a r i e d b e t w e e n 3 2 .4 a n d 3 9 .5 ~ A p l o t o f t h e a c t u a l r a t e o f o x y g e nc o n s u m p t i o n (Q o..) v e r s u s t h e s t o ie h i o m e t r i c a m o u n t o f o x y g e n r e q u i re df o r c o m p l e t e o x i d a t i o n o f t h e s u b s t r a t e c o n s u m e d (k 2) i s s h o w n i n F i g . 8 .T h e d a t a f i t a c u r v e Qo~ : b x (21a)where b = s lope = f ract ion of subs tra te oxidized; x = oxygen required for com-ple te oxidat ion of subs tra te consumed.U s i n g t h e l e a s t s q u a r e s m e t h o d t h e s l o p e w a s f o u n d t o b e 0 . 3 7 . T h i sm e a n s t h a t i n d e p e n d e n t o f t h e g r o w t h r a t e 3 7 ~ o f t h e g l u c o se a s s i m i l a te dw e r e a c t u a l l y o x i d i z e d i n t h e m e t a b o l i c p r o c e s s .


12/20

1 / / 0 0

K A R L L S C H U L Z E a n d R O B E R T S . L I ]? E :

5 0O,h 50O

~ J00

oe

T h e d a t a f o r k a l i s t e d i n c o l u m n 6 o f T a b l e 1 c a n a l s o b e u s e d t o e s t a b -l is h a r e l a t io n s h i p b e t w e e n g r o w t h r a t e a n d t h e r a t e o f s u b s t r a t e u p t a k e .A p l o t o f / c a v e r s u s D ~ k 1 i s s h o w n i n F i g . 9 . T h e c u r v e d e m o n s t r a t e s t h a tk S i n c r e a s e s in d i r e c t p r o p o r t i o n t o t h e g r o w t h r a t e f o l lo w i n g t h e e q u a t i o n

k , = n + h k 1 (22)where n ----y - i n t e r c e p t , r e p r e s e n t i n g g r a m s o f g lu c o s e u p t a k e p e r g r a m c e i l w e i g h tpe r ho ur a t k 1 -- --0.

_ o

o o

o c o

200 tl 00 800 800 10 00 1200O ~ r e p u l r e o or eom f le te ox idc lt /on / 1T ~I ]mr

: F i g 0 R e la t io n s h ip b e tw e e n a c tu a l r a te o f o x y g e nu p t a k e a n d t h e s t o ie h i o m e t r ie a m o u n t o f o x y g e nr e q u i r e d f o r c o m p l e t e o x i d a t i o n o f t h e s u b s t r a t ea s s i m i l a t e d . S l o p e = 0 . 8 7F i g . 9 . R e l a t i o n s h ip b e t w e e n g r o w t h r a t e k l a n d r a t eo f s u b s t r a t e u p t a k e k 2

. ~ I Z O 0

I 0 0 0

~ 8 0 0

6 0 0

~ 0 0

200

2

1 ~ I ~ I0 2 a / / 0 8D = / ~ ~ k r jF i g . 9

h - ~ c o n s t a n t , r e p r e s e n t i n g g r a m s g l u c o se c o n s u m e d p e r g r a m c e l lw e i g h t fo r m e d . U s i n g t h e l e a s t s q u a r e s m e t h o d v a l u e s f or n = 0 .0 5 5 a n dh -= 1.9 1 w e r e o b t a i n e d s o t h a t

kS = 0.055 -~ 1.91 k 1 (23)T h e c o n s t a n t h r e p r e s e n t s t h e i n v e r s e o f a y i e l d c o n s t a n t Y w h i c h h a s b e e nc o r r e c t e d f o r m a i n t e n a n c e s u b s t r a t e c o n s u m p t i o n . A c c o r d i n g t o t h is c o n -c e p t t h e a n a l y s i s o f t h e c u r v e in F i g . 9 p r o d u c e s a v a l u e o f Y = 1 / h = 0 . 5 2w h i c h i s c o n s t a n t o v e r t h e f u l l r a n g e o f g r o w t h r a t e s . T h e s p ec i f ic r a t e o fs u b s t r a t e r e m o v a l c a n t h e r e fo r e b e e x p r e s s e d a s

k 2 = n + k U Y (24)a n d t h e g r o w t h r a t e c a n b e r e l a t e d to t h e r a t e o f s u b s t r a t e r e m o v a l b y

k 1 = Y (k~ - - n) (25)


13/20

Substrate concentration growth and respiration rate in continuous culture 13By substituting equation (20) into equation (24) the rate of substrateremoval can also be related to the rate of oxygen consumption as follows :

Qo~- 16/ ~ = n - ~ 770Y' (26)Furthermore since /c1 is related to the substrate concentrati on by equa-tion (8) or (10) the substrate removal rate can now be expressed in termsof substrate concentration by

k e = n d - : Y ' \ S ~ d - S / (27)or by

4 = n ~ (1 - e-c~) (28)YThese equations would allow to predict the substrate removal rate at agiven substrate concentration if the constants n , Y ' , t c ~ and S~ or c areknown. Thus for n = 0.055, k~ = 0.76, and Y' = 0.52 a ma ximu m valuefor the substrat e r emov al rate of k s = 1.51 g glucose per gram cell weightper hour is obtained from equation (24) which is in agreement with thevalues actually measured as shown in Table 1, column 6. These aspectsof microbial activ ity are very impor tan t in the waste trea tme nt field andwill be discussed in a separate publication.In the development of continuous flow equations the growth rate isusually assumed to be a constant fraction of the substrat e consumptionrate

d x / d t = - - Y d s / d t (29)and inversely the rate of substrate consumption is given by

d s / d t = - - d x / Y d t = - - ] c l x /Y (30)Now it appears th at equation (30) has to be modified by a small amo untof substrate consumption which is evident even if no growth occurs. Inthe case described here this amou nt would be n = 55 mg glucose per gramcell weight per hour so that

d s / d t = - - ( n x ~ - I ~ l x / Y ' ) (31)and

d s / x d t = k 2 = - - ( n ~ - k l / Y ' ) (31 a)which is identical ~dth equat ion (24) developed from the curve in Fig. 9.This means that at a substrate removal rate of/c 2 = 0.055, k 1 would bezero. It appears th at this is an expression for the mainte nan ce supplyof bacterial cells since the consumption of 55 mg glucose per gram cellweight per hour would be just enoug h to sustain the cell metabolism butnot enough to allow growth and reproduction. These conclusions receivedadditional suppor t from the investigation of the b ehavio ur of the conti-nuous flow culture at very low feed rates.


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14 KArL L. SC~ULZ]~ an d ~:~OBERT S LIPE:C o n t i n u o u s l o w c u l tu r e a t l o w D v a l u e s

Fro m equat ion (5) it follows tha t the cell concentra tion in the reactorshould reach its max im um value as D approaches zero : x = Y S .

In experiment No. 3 the continuous flow unit was operated at decreas-ing feed rates so that the D values ranged from 0.081 to 0.011 per hour.The results are listed in Table 3. The data indicate tha t steady state con-ditions were maintained down to a D value of 0.02 which corresponds toa k 1 value of 0.02, a generat ion time of In 2 k 1 = 34.6 hours and a reten-tion time of l I D = 50 hrs. As expected the subst rate concentr ation in the

Table 3. Experiment No 3Continuous flow culture at low D values. Avg. substrate conc. in feed 966 rag/1

glucose. Avg. pI~ in r eact or 6.6days Dlhr rate of substrate supplyof operation mg/g cell wt/hr1 st3rd5th7th

12th17th22 nd31 st37 th

0.0810.0620.0520.020.020.0110.0110.0110.011

cell conc.mg/1 rag/1438 6.2422 5.0442 4.1422 O 1434 O 13 8 0 0 1355 0 1314 0 1293 0 1

178142116 steady

s t a t e64 5

2830 nonsteady34 state36

reactor decreased to near zero. The unit was operated for 5 days underthese conditions. When the feed rate was decreased to D = 0.011, the eel1concentra tion began to drop off to 380 rag/1 after 5 days and to 293 mg/lafter 25 days of operation. Ap pare ntl y under these conditions the rate ofloss of cells was greater tha n the rate of reprod uction and k~ must havedecreased to a value near zero. The rate o f subst rate supp ly in mg glucoseper gram cell weight per hour as found by D S 103/2 is given in column 5of Table 3. According to these dat a stead y state operation at least for5 days was possible down to a supply r ate of 45 nag glucose per gram cellweight per hour. When the supply rate decreased to 28 mg glucose pergra m cell weight per hour at D =- 0.011 the g rowth rate appr oach ed zero.The supply rate in this case can be set equal to the removal rate sincepractically no substrate was lei~ in the effluent. App aren tly under theseconditions the rate of substrate supply was not sufficient to allow cellreproduc tion even at a rate of k 1 = 0.011. Going back to the resultsobtained from a plot of k 1 versus k 2 in Fig. 9 it will be not ed th at a basicsubst rate remova l rate of n = 55 mg glucose per g ram cell weight perhour was indicated at k 1 = 0. tte re in exper iment No. 3 the loss of steady


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Sub stra te concen tra t ion grow th and respira t ion ra te in continuous cul ture 15s t a t e c o n d i t i o n s o c c u r r e d a s s o o n a s l e ss t h a n 4 5 m g g lu c o s e p e r g r a m c e llw e i g h t p e r h o u r w e r e c o n su m e d . T h u s t h e se d a t a s u p p o r t t h e i d e a t h a ta c e r ta i n m i n i m u m c o n s u m p t i o n r a t e o f t h e e n e r g y s u p p l y i n g s u b s t r a t ei s n e e d e d j u s t t o m a i n t a i n t h e c ells a n d t h a t a c t i v e g r o w t h b e g i n s a tc o n s u m p t i o n r a t e s a b o v e t h i s l ev e l. E v i d e n t l y a d d i t i o n a l e x p e r i m e n t sw o u l d b e n e e d e d t o d e t e r m i n e t h e e x a c t v a l u e o f n , i .e . w h e t h e r i t w o u l db e c l o se r t o 5 5 o r t o 4 5 m g g l u c o se p e r g r a m c e ll w e i g h t p e r h o u r . R e c e n t l yM A R R, N I L SO ~ a n d C LA ~K ( 19 6 3) i n v e s t i g a t e d t h e m a i n t e n a n c e r e q u i r e -m e n t o f E . c o l i b y u s i n g a s o m e w h a t d i f f e r e n t m o d i f i c a t i o n o f e q u a t i o n (2 9) :

d x / d t + a x = Y d s / d t (32)w h e r e a h a s t h e d i m e n s i o n o f r e c i p r o c a l t i m e a n d is d e s i g n a t e d a s t h es p e ci fi c m a i n t e n a n c e r a t e . D i v i d i n g e q u a t i o n ( 32 ) b y x y i e l d s :

d x / x d t + a = Y d s / x d t (33)S i n c e d x / x d t = k I a n d d s / x d t ---- k 2 w e o b t a i n

a = Y 'k2 - - k1 (36)F r o m e q u a t i o n (3 1 a ) :n = k z - - k ~ / Y = a / Y (35)

T h u s it t u r n s o u t t h a t o u r s p ec if ic m a i n t e n a n c e c o n s u m p t i o n r a t e n a n dt h e s pe c if ic m a i n t e n a n c e r a t e a c a n b e e a si ly c o n v e r t e d i n t o e a c h o t h e r b yt h e y i e l d c o e f fi c ie n t Y ' . A s s u m i n g Y ' t o b e 0 .5 2 w e o b t a i n a = - Y ' n - - 0 . 0 2 8f o r t h e s t r a i n o f E . c o l i u s e d in o u r e x p e r i m e n t s . M A ~ e t al . (1 9 63 ) l i s te da ~ - 0 . 0 1 8 fo r E . c o l i s t r a i n M L 3 0 a n d 0 .0 2 8 f o r s t r a i n P S . T h e a g r e e m e n tb e t w e e n t h e s e v a l u e s i s s u r p r i s i n g l y c l o s e . A p p a r e n t l y a r e p r e s e n t s t h em i n i m u m v a l u e f o r t h e d i l u ti o n r a t e D a t w h i c h s t e a d y s t a t e is p o s si b le i na g i v e n c a se . R e g a r d l e s s o f t h e s p e ci fi c v a l u e o f n w h i c h m a y b e o b t a i n e di n a n y i n d i v i d u a l c as e, t h e e x p e r i m e n t a l d a t a s h o w t h a t e q u a t i o n (5) i sv a l i d o n l y b e t w e e n c e r t a i n l im i t s o f D , i n t h e c a se d e s c ri b e d h e r e b e t w e e nD ~ 0 . 0 2 a n d D e ~ 0 . 7 6 .

A p p l i c a t i o n o [ s t e a d y s t a t e e q u a t i o n s to e x p e r i m e n t a l d a t aA s m e n t i o n e d p r e v i o u s l y t h e d e v e l o p m e n t o f e q u a t i o n s (5) a n d (6)

d e f in i n g s t e a d y s t a t e c ell- a n d s u b s t r a t e c o n c e n t r a t i o n s w a s b a s e d o ne q u a t i o n ( 8 ) . S i n c e i t w a s f o u n d t h a t e q u a t i o n ( 1 0 ) f i t s t h e r e l a t i o n s h i pb e t w e e n k 1 a n d s u b s t r a t e c o n c e n t r a t i o n b e t t e r , n e w e q u a t i o n s f o r 2 a n d ~ 'w e r e d e v e l o p e d u s i n g e q u a t i o n ( 1 0 ) a s a b a s i s . S o l v i n g e q u a t i o n ( 1 0 ) f o rS y i e l d s : k~

? (361a n d s u b s t i t u t i n g e q u a t i o n (3 6) f o r S i n e q u a t i o n (5 ) r e s u l t s in :[ 1 ~2 = Y S' ~ (37)


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1 6 K A R L L . S C ~ V L Z E a n d I%OB~Rm . L I P ~ :These equations should define the substr ate concentra tion and the cell

concentrat ion in the reactor or effluent for any D value between 0.02 andD = /c ~ ff the constants ]ca, Y, and c are known. I n Figs. 10 and 11 thetheoretical curves for S and 2 are compared with the correspondingexperimental data from run No. 1.

The curves demonstrate that the equations predict the substrate con-centration in the effluent and the cell concentr ation in the reactor in good

800

6 0 0

~/dOgI ZOO

2 Oq 8 ~ 8Fig, 10

. % . r , I : .F i g . 1 1

F i g . 1 0 . C o m p a r i s o n o f S v a l u e s , c o m p u t e d f r o m e q u a t i o n 3 6 ) ; o e x p e r i m e n t a l d a t aF i g . 1 1. C o m p a r i s o n o f s v a l u e s . - - c o m p u t c d f r o m e q u a t i o n 3 7 ); . . . . . c o r r e c te d f o r m a i n t e n a n c e

r e q u i r e m e n t , e q u a t i o n 4 2 ); o e x p e r i m e n t a l d a t aagreement with the experimental results. The agreement between experi-mental and theoretical substrate concentration is especially close. Thediscrepancies between the theoretical and the experimental cell concen-trations are probably related to the fact that in equation (37) an averagevalue for the yield factor was used, whereas, actually the yield factordecreased with decreasing D values.

I f in addition to eq uati on (10) equa tion (31) is also accept ed thiswould necessitate a modification of equa tion (37). The balance equat ion iorthe net rate of change of substrate concentratio n in the reactor used so farw a s increase = input - - output -- consumption

d s / d t ~ D S - - D S - - k l x / Y (38)This would now be modified by the minimum consumption factor n:

d s / d t : D S - - D S - - x ( n - ~ k l / Y ' ) (39)For steady state conditions where d s l d t = 0 we obtain

D ( s - ~ ) = ~ ( ~ + 4 / Y ) ( 4 0 )Solving for 2 and introduc ing k 1 = D yields

: Y ( s , - ~ )-- n Y (41)D ~ - 1


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Sub stra te concentra t ion grow th and respira t ion ra te in continuous cul ture 17S u b s t i t u t i n g e q u a t i o n ( 3 6 ) i n t o ( 4 1 ) p r o d u c e s

Y' S ' k~ D _= n Y" (42)

D + 1T h e e q u a t i o n i s i d e n t i c a l w i t h e q u a t i o n (3 7) e x c e p t fo r t h e c o r r e c t i o n

f a c t o r (n Y ' /D) @ 1 . U s i n g t h e c o n s t a n t s e v a l u a t e d p r e v i o u s l y n Y ' = 0 . 0 2 8 .F o r h i g h D v a l u e s t h e c o r r ec t io n w o u l d b e s m a l l w h e r e a s a t l o w D v a l u e st h e c o r r e c t i o n f a c t o r w o u l d d e c r e a s e t h e c e ll c o n c e n t r a t i o n s i g n i fi c a n t ly .T h i s is i n g e n e r a l a g r e e m e n t w i t h t h e e x p e r i m e n t a l d a t a . U s i n g t h e s a m ev a l u e s f o r Y', 1 a n d c , ~5 w a s c o m p u t e d a c c o r d i n g t o e q u a t i o n (4 2) a n dt h e r e s u l t i n g c u r v e p l o t t e d a s a d o t t e d l in e i n F i g . 1 1. T h e g r a p h d e m o n -s t r a t e s t h a t t h e i n t r o d u c t i o n o f t h e c o r r ec t io n f a c t o r f o r m a i n t e n a n c es u b s t r a t e c o n s u m p t i o n p r o d u c e d a n i m p r o v e d a g r e e m e n t b e tw e e n t h et h e o r e t i c a l a n d t h e e x p e r i m e n t a l c e ll c o n c e n t r a t i o n .

T h e a n a l y si s o f t h e c ell m a t e r i a l g r o w n i n t h e r e a c t o r s h o w e d t h a t t h ea v e r a g e c e l l d e n s i t y o f 4 2 2 m g /1 d r y w e i g h t c o r r e s p o n d e d t o 7 l 0 s v i a b l ec e ll s p e r m l . T h e n i t ro g e n c o n t e n t w a s 12 .1 ~ o f d r y w e i g h t a n d t h e a s ha m o u n t e d t o 8 .5 ~ o f d r y w e i g ht .

i s cus s ionG r a p h i c a l a n d n u m e r i c a l a n a ly s is d e m o n s t r a t e d t h a t t h e a p p l i c a t io n

o f t h e T e i ss ie r e q u a t i o n r e s u l te d i n c lo se a g r e e m e n t w i t h t h e e x p e r i m e n t a ld a t a , w h e r e a s , t h e a p p l i c a t i o n o f t h e 5 i ch a e li s- ? v[ en t en e q u a t i o n p r o d u c e dv a l u e s f o r t h e c o n s t a n t s k ~ a n d S ~ w h i c h c o u l d n o t b e v e r if ie d . U s u a l l y i ti s a s s u m e d t h a t S n, t h e s u b s t r a t e c o n c e n t r a t i o n a t w h i c h t h e g r o w t h r a t er e a c h e s k ~ / 2 h a s a v e r y l o w v a l u e . M o ~-o D ( 19 4 2) o b t a i n e d S n = 2 , 4 a n d2 0 m g / l fo r m a n n i t o l , g l u co s e a n d l a c t o se r e s p e c t iv e l y a n d t t E ~ E R T e t a ] .( 19 5 6 ) c o m p u t e d S ~ = 1 2 .3 m g /1 f o r g l y c e r o l a s l i m i t i n g s u b s t r a t e . I f S isc o m p u t e d f r o m t h e n e w l y d e v e l o p e d e q u a t i o n ( 36) f o r D = k 1 = k~/2a v a l u e o f 4 9 ra g/1 i s o b t a i n e d w h i c h i s i n a g r e e m e n t w i t h t h e e x p e r i m e n -t a l d a t a . A p p a r e n t l y t h e v a l u e o f S ~ d e p e n d s o n t h e t y p e o f s u b s t r a t ew h i c h c o n s t i tu t e s t h e l i m i ti n g g r o w t h f a c t o r a n d o n t h e t y p e o f o r g a n i s mw h i c h i s c u l t i v a t e d .

E q u a t i o n (36) c a n a ls o b e u s e d t o e s t i m a t e t h e s u b s t r a t e c o n c e n t r a -t io n a t w h i ch t h e m a x i m u m g r o w t h r a t e is a t t a i n e d a n d a b o v e w h i ch t h eg r o w t h ra t e b e c o m e s i n d e p e n d e n t o f t h e s u b s t r a t e c o n c e n t r a t io n . F o rD = k ~ t h e s u b s t r a t e c o n c e n t r a t i o n w o u l d b e i n fi n it e . ] { o w e v e r , f o rp r a c t i c a l p u r p o s e s i t w o u l d b e s u f f i c i e n t t o o b t a i n t h e s u b s t r a t e c o n c e n -t r a t i o n a t w h i c h k 1 r e a ch e s 9 5 ~ o f i t s m a x i m u m v a h i e . T h e r e s u l t in gv a l u e o f S i s 2 1 4 m g /1 g l u c o s e w h i c h i s i n a g r e e m e n t w i t h t h e e x p e r i m e n -t a l d a t a .

A r c h . ~ { i k r o b i o l . ] ~ d . 4 8 2


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18 KARL L . SC~VLZEand ROBERTS. LI~E:B a s e d o n e x p e r i m e n t a l w o r k b y M o N o I ) ( 19 42 ) i t is u s u a l l y a s s u m e d

t h a t t h e y i e l d f a c t o r is c o n s t a n t a n d i n d e p e n d e n t o f t h e g r o w t h r a t e . I t isn o t e w o r t h y , h o w e v e r , t h a t a ll t h e y i e l d f a c t o r s d e t e r m i n e d b y M ON ODw ere r e l a t i v e ly l o w, i . e . Y ---- 0 . 2 3 fo r E col i w i t h g l u c o s e a s l im i t i n g s u b -s t r a t e . I n o t h e r w o r d s, 4 . 35 g o f g l uc o s e w e r e c o n s u m e d i n t h e f o r m a t i o no f 1 g c e ll m a t e r i a l w h e r e a s , i n t h e c a se r e p o r t e d h e r e o n l y a n a v e r a g e o f1 .9 3 g o f g l u c o se w e r e c o n s u m e d i n t h e f o r m a t i o n o f I g c e ll m a t e r i a l . I tis p o ss ib l e, th e r e f o r e , t h a t t h e n u t r i e n t m e d i u m u s e d b y ~ O ~ O D w a s in -c o m p l e t e a n d t h a t o t h e r l i m i t i n g f a c t o r s b e s i d e s t h e c a r b o h y d r a t e w e r ep r e s e n t .

T h e f a c t t h a t i n d e p e n d e n t o f t h e g r o w t h r a t e o n l y 37 ~ o f t h e g l u co s ec o n s u m e d w e r e o x i d i ze d is in a g r e e m e n t w i t h d a t a r e p o r t e d b y P o ~ G E se t a l. ( 1 95 6) . T h e s e a u t h o r s f o u n d i n W a r b u r g e x p e r i m e n t s w i t h m i x e db a c t e r i a l c u lt u r e s a n d s k i m m i l k a s a s u b s t r a t e t h a t a t t h e p o i n t o f c o m -p l e te s u b s t r a te r e m o v a l 37 ~ o f t h e t h e o r e t ic a l o x y g e n d e m a n d h a d b e e ne x e r t e d . B a s e d o n t h i s a n d o n a s i m p l i f i e d e m p i r i c a l m o l e c u l a r f o r m u l af o r b a c t e r i a l c e ll s u b s t a n c e t h e y d e v e l o p e d t h e f o ll o w i ng e q u a t i o n f o r t h eb a c t e r i a l g r o w t h p r o c e s s :

8 (Ctt~O) -~ 30 2 -~ NH ~-+ C~HTNO~~- 3C02 -~ 6H20A c c o r d i n g t o t h i s e q u a t i o n i n t h e c o n s u m p t i o n o f 1 00 g c a r b o h y d r a t e 4 0 g0 2 a r e r e s p i r e d a n d 4 7 . 1 g c e ll m a t e r i a l a r e p r o d u c e d . T h i s c o r r e s p o n d st o 3/s o r 3 7.5 0/0 o f t h e a m o u n t o f o x y g e n r e q u i r e d f o r c o m p l e t e o x i d a t i o no f t h e c a r b o h y d r a t e c o n s u m e d a n d t o a y i e l d f a c t o r o f Y ~ - 0 .4 7 1. S i n cet h e e m p i r i c a l f o r m u l a r e p r e s e n t s o r g a n i c ce ll m a t e r i a l w i t h o u t a s h t h ey i e l d f a c t o r w o u l d h a v e t o b e i n c r e a s e d t o Y - ~ 0.5 1 f f a n a s h c o n t e n t o f8 .5 0/0 i s c o n s i d e r e d . T h i s a g r e e s w e l l w i t h t h e y i e l d c o n s t a n t o f 0 .5 2o b t a i n e d f r o m g r a p h i c a l a n a l y s is o f s u b s t r a t e u p t a k e d a t a .

I n c o n n e c t io n w i t h t h e W a r b u r g m e a s u r e m e n t s c a l cu l a ti o n s hb w s t h a tt h e s u p p l y o f g lu c o se p r e s e n t i n t h e s a m p l e s t a k e n f r o m t h e r e a c t o r sh o u l db e e x h a u s t e d i n 5 t o 1 3 r a in a t t h e p r e v a i li n g r a t e s o f s u b s t r a t e r e m o v a l .T h e r e is n o d o u b t t h a t i n t im e t h e r e s p i r a t i o n r a t e s o f t h e s e c e ll s w o u l dh a v e d e c li n ed t o w a r d t h e e n d o g e n o u s le v el . H o w e v e r t h e W a r b u r g d a t as h o w e d t h a t n o s i g n if i ca n t d e c r e a se o f t h e r e s p i r a t i o n r a t e s o c c u r r e d i nt h e 1 5 t o 3 0 r a i n p e r i o d d u r i n g w h i c h r e a d i n g s w e r e t a k e n . T h i s l e a d s t ot h e c o n c l u si o n t h a t a c e r t a i n a m o u n t o f i n t r a c e l l u la r s t o r a g e o f s u b s t r a t em u s t b e i n v o l v e d i n t h e r e s p i r a t o r y p r oc e s s.

W i t h r e g a r d t o t h e c e s s a ti o n o f c e ll r e p r o d u c t i o n a t v e r y l o w D v a l u e so n l y f e w s im i l a r o b s e r v a t i o n s h a v e b e e n r e p o r t e d i n t h e l i t e r a t u r e ( H E ~ -BERT 1958 b ; N OVICK 1958 ; JANNASCH 1963). OVICK 1958) m e n t i o n e dt h a t n o s t e a d y s t a t e w a s p o s s i b l e w h e n c u l t i v a t i n g a t r y p t o p h a n er e q u ir in g m u t a n t o f E col i a t D v a l u e s b e l o w 0 . 0 3 . A s a n e x p l a n a t i o n ,N o v i c e : a s s u m e d t h a t t h e b a c t e r i a w e n t i n t o a k i n d o f l a g st a t e . M o N oD


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Sub stra te concentra t ion grow th and respira t ion ra te in continuous cul ture 19( 1 9 4 2 ) h a d p o s t u l a t e d t h a t t h e o r e t i c a l l y a m i n i m u m s u b s t r a t e c o n c e n -t r a t i o n ( c o n c e n t r a ti o n d e n t r e t i e n ) s h o u l d e x i s t w h i c h w o u l d a l l o w o n l ym a i n t e n a n c e o f t h e c e lls . H o w e v e r , s u ch a m i n i m u m s u b s t r a t e c o n c e n -t r a t io n c o u ld n o t b e p r o v e n e x p e r i m e n t a ll y . N o w i t a p p e a r s t h a t n o t t h es u b s t r a t e c o n c e n t r a t i o n a s s u c h i s t h e c r it ic a l p o i n t b u t t h e m i n i m u m r a t eo f s u b s t r a t e u p t a k e b e l o w w h i ch t h e c ells w o u l d n o t b e c a p a b l e o f r e p r o -d u c t i o n .

SummaryU s i n g a c o n t i n u o u s f lo w t e c h n i q u e t h e r e l a ti o n s h i p b e t w e e n g r o w t h

r a t e a n d s u b s t r a t e c o n c e n t r a t i o n w a s i n v e s t i g a t e d w i t h g l u c o se as th el i m i t i n g f a c t o r o f a c u l t u r e o f E s c h e r i c h i a c o l i G r a p h i c a l a n d n u m e r i c a la n a ly s is o f t h e e x p e r i m e n t a l d a t a d e m o n s t r a t e d t h a t t h e a p p l i c a t i o n o ft h e N i c h a e h s - M e n t e n e q u a t i o n p r o d u c e d e r r o n e o u s r e s u l t s , w h e r e a s , t h ec o n s t a n t s o b t a i n e d f r o m t h e T e i s si e r e q u a t i o n w e r e i n a g r e e m e n t w i t ht h e e x p e r i m e n t a l d a t a . O n th i s b a si s, n e w e q u a t i o n s d e f in i ng t h e s t e a d ys t a t e c e ll a n d s u b s t r a t e c o n c e n t r a t i o n i n c o n t i n u o u s f lo w c u l t u r e s w e r ed e v e l o p e d a n d t e s t e d a g a i n s t e x p e r i m e n t a l d a t a .

C o m p a r i s o n o f t h e s p e ci fi c g r o w t h r a t e s, s u b s t r a t e u p t a k e r a t e s a n do x y g e n c o n s u m p t i o n r a t e s d e m o n s t r a t e d t h a t a ll w e re d i re c t ly p r o p o r -t io n a l t o e a c h o t h e r a n d c o u ld b e re l a te d t o e a c h o t h e r b y m a t h e m a t i c a le q u a t i o n s . S p e c i fi c a ll y i t w a s s h o w n t h a t a s t h e g r o w t h r a t e i n c r e a s e df r o m 0 .0 6 t o k ~ = 0 .7 6 t h e s u b s t r a t e u p t a k e r a t e i n c r e as e d f r o m 1 3 4 t o1 42 0 m g g l u co s e p e r g r a m c el l w e i g h t p e r h o u r a n d t h e o x y g e n c o n s u m p -t i o n r a t e i n c r e a s e d f r o m 4 8 .6 t o 5 05 m g 0 2 p e r g r a m c el l w e i g h t p e r h o u r .I n d e p e n d e n t o f t h e g r o w t h r a t e 37 ~ o f t h e c a r b o h y d r a t e c o n s u m e d w e reo x i d i ze d . T h e y i e l d f a c t o r v a r i e d f r o m 0 . 4 4 a t l o w g r o w t h r a t e s t o 0 .5 4 a th i g h g r o w t h r a t e s. A n a l y s is o f t h e g r o w t h r a t e - s u b s t r a t e u p t a k e r a t er e la t io n s h ip i n d i c a te d t h a t a m i n i m u m s u b s t r a t e u p t a k e r a t e o f 5 5 m gg l u c o s e p e r g r a m c e ll w e i g h t p e r h o u r e x i s t e d b e l o w w h i c h c el l r e p r o d u c -t i o n w o u l d c e as e. T h i s w a s s u p p o r t e d b y t h e f a c t t h a t s t e a d y s t a t e c o n -d i t i o n s c o u l d no t, b e m a i n t a i n e d i n t h e c u l t u r e a t D v a l u e s b e l o w 0 .0 2w h e n t h e s u b s t r a t e s u p p l y r a t e d e c r e a se d b e l o w 45 m g g l u co s e p e r g r a mc e l l w e i g h t p e r h o u r .

eferencesD ixo n, M . , and C. E. W ~BB: Enzym es, p . 21. Academic P ress Inc . (1958).HE~BnUT, D. : Continuous cultu re o f microorganisms; some theo retical aspects. I n :Continuous Cultivation of Microorganisms; A Symposium, p . 45 --52 . Pra gu e:

Czechoslovak A cad em y o f Sciences (1958 a) .-- VII . Int. Congr. Microbiol. Syrup. 381--396, Stockholm (1958b).-- g. ELSWORTU,and R. TELLING:The cont inuous cul ture of bacte r ia ; a theoret ica land e xp erim enta l stud y. J . gen. Microbiol. 15, 601--622 (1956).HOELSC~R, 1~ ., J . A ~0 LD , a nd H . P i ra cy : Graph ic a ids in engineering compu-ta t ion, p . 3 6 . N e w Y o r k : M c G r a w - H i l l 1 9 5 2 .2


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Escherichia col i Ann. N.Y. Acad. Sci. 102, 536 1963).Moron, J. : Recherches sur la croissance des cultures baeteriennes. 210 pp. Par is:Her mann and Cie. 1942.The growth of bacterial cultures. Ann. Rev. Mierobiol. 3, 371--394 1949).La technique de culture continue; Th6orie et applications. Am1. Inst. Pasteur 79,390--410 1950).MOOl~E, E., H. T~olviAs, and W. S~ow: Simplified method for analys is of B.O.D.data. Sewage Ind. Wastes 22, 1343--1353 1950).:NELson, N. : A photometric adap tat ion of the Somogyi method for the determina-tion of glucose. J. biol. Chem. 153, 375--380 1944).:NOVlCX, A. : Growth of bacteria. Ann. Rev. Microbiol. 9, 97--110 1955).-- Genetic and physiological studies with the chemostat. In : Symposium on conti-nuous cultivat ion of microorganisms, p. 29. Prague: Czechoslovak Academy ofSciences 1958.-- , and L. SZILARD: Experiments with the chemostat on spontaneous mutati on ofbacteria. Proe. nat. Acad. Sci. Wash.) 36, 708--719 1950).Official methods of analysis. 7th ed., 910 pp. Washington 4, D.C. : Assoc. of OfficialAgricultural Chemists 1950.POl~GEs, N., L. JASEWlCZ, and S. Hoovm~: Principles of biological oxidation. I n:Biological treatment of sewage and industr ial wastes. Vol I., p. 35--48. NewYork: Reinhold Publ . Co. 1956.:REED, L. J., and E. J. T]t~XlAIILT: The statist ical t rea tme nt of react ion velocitydata. g. Phys. Chem. 35, 950--971 1931).;ScgvLZ~, K. L. : The effect of phosphate supply on the rate of growth and fa~ for-mation in yeasf0s. App]. Microbiol. 4, 207--210 1956).;SPEcTOl~, W. S. : Handbook of biological data. Phil., Pa. : W. ]3. Saunders Co. 1956.~Standard Methods for the Examination of Water, Sewage and Industri al Wastes,10Lh Ed. New York: Amer. Publ . Health Assoc., Inc. 1955.T]~ISSI]m, G.: Les lois quanti tatives de la croissance. Ann. Physiol. physieochim.biol. 12, 527--586 1936).T~o~As, H. H.: The slope method of evaluating the constants of the first stagebiochemical oxygen demand curve. Sewage Works J. 9, 425--430 1937).U~BI~IT, W., R ]3tmt~IS, and J. STAVSSm~: Manometric Techniques. 338 pp.Minneapolis, Minn. : Burgess Publ. Co. 1957.

Dr. K. L. Sc~uLz]~, Associate Professor,Division of Engineering Research, Michigan State University,East Lansing, Michigan, U.S.A.Dr. R. S. L1TE, Assistant Professor,Science Department, Southwest Missouri State College,Springfie]d, Missouri, U.S.A.

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1964 AFM Relationship Between Substrate Concentration, Growth Rate and Respiration Rate of E. Coli in Continuous Culture